Impurities in Solids

8/9/2019 Impurities in Solids

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Physics 243 - Lecture Notes

Chetan Nayak

January 27, 2004


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Contents

1 Impurities in Solids 1

1.1 Impurity States . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Density of states*** . . . . . . . . . . . . . . . . . . . . . . . 31.3 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Anderson Model . . . . . . . . . . . . . . . . . . . . . 31.3.2 Locator expansion*** . . . . . . . . . . . . . . . . . . 51.3.3 Anderson Insulators vs. Mott Insulators . . . . . . . . 5

1.4 Physics of the Insulating State . . . . . . . . . . . . . . . . . 7

1.4.1 Variable Range Hopping . . . . . . . . . . . . . . . . . 71.4.2 AC Conductivity . . . . . . . . . . . . . . . . . . . . . 91.4.3 Effect of Coulomb Interactions . . . . . . . . . . . . . 101.4.4 Magnetic Properties . . . . . . . . . . . . . . . . . . . 12

1.5 Physics of the Metallic State . . . . . . . . . . . . . . . . . . 161.5.1 Disorder-Averaged Perturbation Theory . . . . . . . . 161.5.2 Lifetime, Mean-Free-Path . . . . . . . . . . . . . . . . 181.5.3 Conductivity . . . . . . . . . . . . . . . . . . . . . . . 201.5.4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 231.5.5 Weak Localization . . . . . . . . . . . . . . . . . . . . 281.5.6 Quasiclassical Approach to Weak Localization*** . . . 33

1.5.7 Weak Magnetic Fields and Spin-orbit Interactions: theUnitary and Symplectic Ensembles . . . . . . . . . . . 33

1.6 The Metal-Insulator Transition . . . . . . . . . . . . . . . . . 331.6.1 Percolation . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6.2 Mobility Edge, Minimum Metallic Conductivity . . . . 35

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4 CONTENTS

1.6.3 Scaling Theory . . . . . . . . . . . . . . . . . . . . . . 37


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CHAPTER 1

Impurities in Solids

1.1 Impurity States

In the previous parts of this book, we have discussed the low-energy excita-tions which result from broken symmetry, criticality, or fractionalization. Inthis final part of the book, we discuss the low-energy excitations which resultfrom the presence of dirt or disorder in a solid. By dirt or disorder,

we mean impurities which are frozen into the solid in some random way.Consider phosphorous impurities in silicon. Presumably, the true groundstate of such a mixture is one in which the phosphorus atoms form a super-lattice within the silicon lattice. However, this equilibrium is never reachedwhen the alloy is made: it is cooled down before equilibrium is reached,and the phosphorus impurities get stuck (at least on time scales which arerelevant for experiments) at random positions. These random static spatialfluctuations have interesting effects on the electronic states of the system:they can engender low-lying excitations and they can dramatically changethe nature of such excitations. To see the significance of this, recall that,in the absence of a broken continuous symmetry, a system will generically

form a gap between the ground state and all excited states. By tuning toa critical state such as a Fermi liquid we can arrange for a system tohave low-energy excitations. However, in the presence of disorder, it willhave low-energy excitations even without any tuning. To get a sense of whythis should be so, suppose that, in the absence of disorder, there were a gap

1


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2 CHAPTER 1. IMPURITIES IN SOLIDS

to the creation of a quasiparticle. In the presence of disorder, the potential

varies from place to place, and, in the thermodynamic limit, there will besome place in the system where the potential energy offsets the energy re-quired to create a qasiparticle. In this region, it wont cost any energy tocreate a quasiparticle. Note that such an excitation, though low in energy,may not have large spatial extent. By the same token, if the system weregapless in the absence of disorder, as in the case of a Fermi liquid, thendisorder can extend the critical state into a stable phase and it can causethe low-lying exciations of the system to become spatially localized.

Consider the simple example of a single phosphorus impurity in silicon.The phosphorus takes the place of one of the silicon atoms. Without thephosphorus, silicon is an insulator with a band gap Eg. Phosphorus has anextra electron compared to silicon and an extra positive charge in its nu-cleus. Let us neglect electron-electron interactions and write Schrodingersequation for the extra electron in the form:

(Hlattice + Himpurity) (r) = E (r) (1.1)

where Hlattice is the Hamiltonian of a perfect lattice of silicon ions andHimpurity is the potential due to the extra proton at the phosphorus site.Let us write (r) in the form

(r) = (r) uk0(r) eik0r (1.2)

where uk0

(r) eik0r is the eigenstate of Hlattice

at the conduction band mini-mum. Then (r) satisfies the equation:

2

2m2 e

2

r

(r) = Eb (r) (1.3)

where m is the effective mass in the conduction band, Eb is measured fromthe bottom of the conduction band, and is the dielectric constant of silicon.

Hence, so long as we can neglect electron-electron interactions, the elec-tron will be trapped in a bound state at the impurity. If the binding energyis much less than the band gap, e

2

2aB Eg, then our neglect of electron-

electron interactions is justified because the interaction will be too weak to

excite electrons across the gap. (Note that aB is the effective Bohr radiusin silicon, aB =

2

me2 20A). Hence, in the presence of impurities, there

are states within the band gap. If there is a random distribution of phos-phorus impurities, we expect a distribution of bound state energies so thatthe gap will be filled in. In other words, there will generically be states at


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1.2. DENSITY OF STATES*** 3

the chemical potential (although there will be a small density of states when

the impurity density is small), unlike in a pure system, where it is possiblefor the chemical potential to lie within an energy gap.

1.2 Density of states***

1.3 Localization

1.3.1 Anderson Model

What is the nature of these electronic states when there are several impu-rities? Presumably, there is some mixing b etween the states at different

impurities. One can imagine that this mixing leads to the formation ofstates which are a superposition of many impurity states so that they ex-tend across the system. As we will see, this naive expectation is not alwayscorrect. Consider, first, the case of a high density of electrons and impurities.One would expect the kinetic energy to increase with the density, n, as n2/d

while the potential energy should increase as n1/d, so that the kinetic energyshould dominate for large n. When the kinetic energy dominates, we mightexpect some kind of one-electron band theory to be valid. This would leadus to expect the system to be metallic. How about the low-density limit? Inthe case of a single impurity, the electron is trapped in a hydrogenic boundstate, as we saw in the previous section. What happens when there is asmall, finite density of impurities? One might expect exponentially small,but finite overlaps between the hydrogenic bound states so that a metal withvery small bandwidth forms. This is not the case in the low-density limit,as we will see in this section.

Consider the Anderson model:

H =i

icici

ij

tijci cj + h.c. (1.4)

In this model, we ignore electron-electron interactions. Spin is an inessentialcomplication, so we take the electrons to be spinless. i is the energy of anelectron at impurity i, and tij is the hopping matrix element from impurity

i to impurity j. These are random variables which are determined by thepositions of the various impurities.

One could imagine such a model arising as an effective description ofthe many-impurity problem. Since the impurities are located at randompositions, there tunneling matrix elements between their respective bound


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states will be random tij. One might also suppose that their different lo-

cations will lead to different effective environments and therefore differentbound state energies i. Anderson simplified the problem by arranging thesites i on a lattice and setting tij = t for nearest neighbor sites and zerootherwise. The effect of the randomness is encapsulated in the is, whichare taken to be independent random variables which are equally likely totake any value i [W/2, W/2]. This is a drastic simplification, but italready contains rich physics, as we will see.

For W = 0, there is no randomness, so the electronic states are simplyBloch waves. The system is metallic. For W t, the Bloch waves areweakly scattered by the random potential. Now consider the opposite limit.For t = 0, all of the eigenstates are localized at individual sites. In otherwords, the eigenstates are

|i, with eigenenergies i. The system is insulating.

Where is the transition between the t/W = and the t/W = 0 limits? Isit at finite t/W = ? The answer is yes, and, as a result, for t/W small,the electronic states are localized, and the system is insulating. (We saythat a single-particle state is localized if it falls off as er/ . is called thelocalization length)

To see why this is true, consider perturbation theory in t/W. The per-turbed eigenstates will, to lowest order, be of the form:

|i +j

t

i j |j (1.5)

Perturbation theory will be valid so long as the second term is small. Sincethe is are random, one can only make probabilistic statements. The typ-ical value of i j is W/2. The typical smallest value for any given i isW/2z, where z is the coordination number of the lattice. Hence, we expectcorrections to be small and, hence, perturbation theory to be valid if2tz/W < 1. On the other hand, this is not a foolproof argument becausethere is always some probability for i j small. Nevertheless, it can beshown (Anderson; Frohlich and Spencer) that perturbation theory convergeswith a probability which approaches 1 in the thermodynamic limit. Hence,there is a regime at low density (small t) where the electronic states arelocalized and the system is insulating, by which we mean that the DC con-

ductivity vanishes at T = 0. From our discussion of the hydrogenic boundstate of an electron at an impurity in a semiconductor, it is not surprisingthat there are localized states in a disordered system. What is surprisingis that if the disorder strength is sufficiently strong, then all states will belocalized, i.e. the entire band will consist of localized states.


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1.3. LOCALIZATION 5

If the disorder strength is weaker than this limit, 2tz/W 1, thenwe can do p erturbation theory in the random potential. In perturbationtheory, the states will be Bloch waves, weakly-scattered by impurities aswe discuss in the next section. Such states are called extended states. Theperturbative analysis is correct for states in the center of the band. Nearthe band edges, however, the perturbative analysis breaks down and statesare localized: the correct measure of the electron kinetic energy, againstwhich the disorder strength should be measured, is the energy relative tothe band edge. Thus, for weak disorder, there are extended states near theband center and localized states near the band edges. When it lies in theregion of extended states, it is metallic.

When the chemical potential lies in the region of localized states, thesystem is insulating. To see this, we write the DC conductance, g, of ssystem of non-interacting fermions of size L in the following form:

g(L) =E

E(1.6)

E is the disorder-avergaged energy change of eigenstates at EF when theboundary conditions are changed from periodic to antiperiodic. E is themean level spacing of eigenstates. (In order to derive such a relation an-alytically, one must consider disorder-averaged quantities.) If the statesat EF are localized with localization length , then E eL/, and theconductance will vanish in the thermodynamic limit.

This form for the conductance can be motivated as follows. The con-ductance is the response of the current to a change in the electromagneticpotential (which can be applied entirely at the boundary). However, thepotential couples to the phase of the wavefunction, so the conductance canbe viewed as the sensitivity of the wavefunction to changes of the boundaryconditions. A systematic derivation proceeds from the Kubo formula.

1.3.2 Locator expansion***

1.3.3 Anderson Insulators vs. Mott Insulators

The preceding analysis shows that a disorder-induced insulating state of

non-interacting fermions is stable over a finite range of hopping strengths.What about the effect of electron-electron interactions? At high density, weexpect the electrons to screen the impurity potentials, so that they are ac-tually of the form e

2

r er. In the high-density limit, , we expect the

impurities to be extremely well-screened so that there wont be any impurity


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1.4. PHYSICS OF THE INSULATING STATE 7

V(r) e2r er has no bound states in the high-density limit, where islarge. Therefore, as the dopant density is decreased, bound states will beginto appear when falls below a certain value aB . This would lead tosmall conductivity. (But not zero yet because the exponentially-small over-lap between these bound states would still be expected to lead to metallicbehavior, albeit with a very narrow band, were it not for localization.) An-derson localization of the impurity bound states will make the crossoverfrom a metal to a poor metal into a sharp metal-insulator transition. Thus,it is more accurate to say that insulating behavior is due to the interplaybetween localization and interactions.

1.4 Physics of the Insulating State

At T > 0, any system will have finite conductivity since there will alwaysbe some probability that thermally-excited carriers can transport chargeacross the system. The conductivity of a finite-sized system will also alwaysbe non-vanishing. For instance, consider a non-interacting Fermi systemwhose single-particle states at the Fermi energy are localized with localiza-tion length . Then the conductivity of the system will be L2d eL/in a system of size L.

Thus, we define an insulator as any state in which the DC conductivityvanishes at T = 0 in the thermodynamic limit. In this section, we discussthe properties of a disorder-driven insulating state.

When a solid is driven into an insulating phase as a result of impurities,it is as far from a perfect crystalline lattice as possible, as far as its elec-tronic properties are concerned. Not only is translational symmetry absent,but physical properties are dominated by localized excitations so that it isdifficult to construct even an averaged description which is translationallyinvariant. However, some of these properties the DC and AC conduc-tivities, magnetic response, and specific heat can be deduced by simplearguments, which we present in this section.

1.4.1 Variable Range Hopping

One of the most celebrated results in the study of disordered insulators is

Motts formula for the DC conductivity due to variable range hopping. Letus consider an insulator at low temperatures. We imagine that the electronsare all in single-particle localized states. However, at finite-temperature,an electron can be excited by, say, a phonon so that it hops to anothernearby localized state. Through a sequence of such hops, it can conduct


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electricity. One might imagine that electrons will only hop to the nearest

neighboring states since the matrix element to hop to a more distant statewill be exponentially (in distance) smaller. However, a state which is furtheraway might be closer in energy, and therefore exponentially (in temperature)easier to reach. The competition between these competing effects leads tovariable range hopping.

Suppose that an electron which is in a localized electronic state hopsto another one a distance R away. The number of available states per unitenergy within such a distance is 43R

3NF, where NF is the density of statesat the Fermi energy. Hence, the typical smallest energy difference betweenthe state and a state within a distance R is the inverse of this:

W(R) typical smallest energy difference betweenstates separated by a distance R = 34NFR3(1.8)

The conductivity associated with such a hopping process depends on thematrix element (squared) between the two states, which we take to be ex-

ponentially decaying e2R (where R = 3R/4 is the average hoppingdistance) and on the probability that the electron receives the activationenergy needed to hop, which follows the Boltzmann distribution eW(R)Hence, the conductivity due to such a hopping process varies as

R

e2RW(R) (1.9)

The prefactor of the exponential contains various comparatively weak de-pendences on R; for instance, there should be a factor of R corresponding tothe fact that a longer hop constitutes a larger contribution to the current.

There are many such hopping processes available to our electron in state. The electron could hop a short distance with relatively large matrixelement, but the activation energy would probably be high in such a case.It could instead hop a long distance and find a state with low activationenergy, but the matrix element would be very small. The optimal route isfor the electron to hop a distance which minimizes the exponential in (1.9):

2 dW

dR = 0 (1.10)

or

Ropt =

3

2NFT

1/4(1.11)


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Hence, as the temperature is decreased, the electron makes longer and longer

hops. The resulting temperature dependence of the conductivity is:

(T) = 0(T) e(T0/T)

1/4

(1.12)

where 0(T) is relatively weakly-dependent on T, e.g. power-law. In ddimensions, the preceeding analysis immediately generalizes to

(T) = 0(T) e(T0/T)

1d+1

(1.13)

1.4.2 AC Conductivity

There are many situations in which the kBT frequency dependence ofa quantity can be obtained from its DC temperature dependence by simplyreplacing kBT with . The conductivity of an insulator does not fall withinthis class. The AC conductivity is entirely different in form.

At frequency , an electron will oscillate between states separated inenergy by . Let us think about these resonant states in terms of hydrogenicbound states. There are two possibilities. (1) R is large, so that they areessentially individual, independent bound states separated in energy by thedifference between their respective potential wells. Then the dipole matrixelement is R eR; the current matrix element is R eR. Since thematrix element will be exponentially suppressed, the contribution of suchresonant pairs will be small, and we turn to the second possibility. (2) Thedistance R is not very large, so that the energy eigenstates are actually linearcombinations of the hydrogenic bound states. Then the energy differencebetween the eigenstates is essentially the matrix element of the Hamiltonianbetween the hydrogenic bound states I0eR. The dipole matrix elementis R; the current matrix element is R. There is no exponentialsuppression because the two eigenstates are different linear combinations ofthe same two bound states.

Then the conductivity due to this resonant pair is:

() (dipole matrix element)2 (available phase space) (1.14)

The available phase space is the density of states per unit volume multipliedby the volume in which the states can lie, given that they are a distance Rapart: N2F R2R. R is the size of a localized state. Hence,

() (R)2 R2 2 R4 (1.15)


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However, by the arguments of the preceding paragraph,

R ln(I0/) (1.16)

Thus, we have the following formula for the conductivity:

() 2 ln4(I0/) (1.17)

This formula also holds at finite temperatures, so long as T. How-ever, when the frequency is decreased, the conductivity will cross over tothe variable-range hopping form.

1.4.3 Effect of Coulomb Interactions

In an insulator, Coulomb interactions are not screened at long-wavelengthsand low-frequencies because there arent mobile charges. Thus, their effectsare particularly strong. One effect, pointed out by Efros and Shklovskii is asuppression of the single-particle density-of-states at the Fermi energy. Thebasic physics is as follows. Suppose that there were two (strongly-localized)single-particle states, one just below the Fermi energy and one just above.The single-particle energy cost associated with promoting an electron fromthe lower one to the higher one would be at least partially offset by thenegative Coulomb energy associated with the resulting particle-hole pair. Ifthe distance between the states is small, this Coulomb energy will be large

and will overcompensate the single-particle energy. Hence, the states mustbe far apart and, therefore, the single-particle density of states (per unitvolume) must be small.

More quantitatively, suppose the single-particle energy separation be-tween an un-occupied state above the Fermi energy and an occupied onebelow the Fermi energy is Eunoccocc while the distance between the statesis r. The single-particle energies include the Coulomb interaction energybetween the two states under consideration and all of the other occupiedstates in the system. Then it must be the case that

Eunoccocc >e2

r(1.18)

or else the energy would be lowered by transferring an electron from theoccupied state to the unoccupied one, contradicting our assumption. Theright-hand-side is the Coulomb energy between the electron and the holewhich results when such a transfer occurs.


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0

EE

N(E)

F

N

Figure 1.1: The single-particle density-of-states is suppressed at the Fermienergy. For |EEF| > , the single-particle density-of-states returns to N0.

Hence, the number of states per unit volume which are within energyE of the Fermi surface is

n(E) 1r3

< (E)3 (1.19)

from which we see that the density of states is

N(E) (E)2 (1.20)

Thus, the single-particle density of states vanishes at the Fermi energy, andis suppressed near it. This effect is called the Coulomb gap.

Suppose we define energy, length scales , r by

e2

r=

N0r3 = 1 (1.21)

where N0 is the density of states in the absence of long-range Coulombinteractions, to which the density of states returns when the distance fromthe Fermi surface exceeds = e3

N0, as shown in figure 1.1.

One could measure the single-particle density of states by performing atunneling experiment in which current tunnels into a disordered insulatorfrom a metallic lead to which it is contacted. The differential conductancedI/dV will be proportional to the density of states. According to the abovearguments, dI/dV will be suppressed at zero voltage and will increase with


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voltage as V2. However, when we change the applied voltage on our disor-

dered insulator, the Fermi energy moves with it. Shouldnt the differentialconductance remain zero? The answer is no. As we change the voltage,dI/dV increases with voltage as V2, as if the chemical potential had notmoved. However, if we raise the temperature of the insulator, change theapplied voltage to V0, and cool it down again, we find that dI/dV variesas (V V0)2, so that the Fermi energy has moved to V0. Essentially, thesuppression of the density of states remains wherever it was when the sys-tem last equilibrated. The non-equilibrium nature of the Coulomb gap isrelated to the assumption above that the only Coulomb interaction energywhich must be accounted for is that between the excited electron and thehole left behind, e2/r. However, when an electron is excited from an oc-cupied state to an unoccupied one, all of the other electrons can and willrearrange themselves. This will occur in such a way as to allow a state nearthe Fermi surface. Thus, if an experiment were done slowly enough to allowsuch re-equilibration, the suppression calculated above would not occur.

The suppression of the density-of states near the Fermi energy has con-sequences for variable-range hopping. Recall that we estimated that within aregion of size R, the typical smallest energy separation Eis E 1/(N0R3).However, N(E) (E)2, or E 1/R, as expected if the energy is de-termined by Coulomb interactions. Following the minimization procedureof subsection 1.4.1, we find that the variable-range-hopping formula in thepresence of Coulomb interactions (in arbitrary dimension) is:

e(T0/T)1/2

(1.22)

1.4.4 Magnetic Properties

In the absence of electron-electron interactions, there will be two electronsin each localized single-particle state, and the ground state will be para-magnetic. However, it is clearly unrealistic to ignore electron-electron in-teractions. We expect them to prevent two electrons from occupying thesame state. Thus, we must consider the spin-spin interactions between theelectrons in different localized states.

An isolated impurity state, occupied by a single electron will have a

Curie form for the magnetic susceptibility:

c(T) 1T

(1.23)

If we have many such spins but ignore the interactions between them, we


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strengths) is that the spins do not order but rather form a random singlet

phase. In such a phase, the system essentially breaks up into a set of pairsof spins which interact with strong antiferromagnetic exchange couplings Ji.Each of these pairs has susceptibility

ithpair(T) =1

T

2

3 + eJi(1.24)

The susceptibility of the system must be summed over all pairs with someprobability distribution, P(J), for the values of J:

(T) =1

T

dJ P(J)

2

3 + eJ(1.25)

At a given temperature, T, there will be some spins whose J is smaller thanT. Such spins will not have formed singlets yet, so they will give a Curie-likecontribution to the susceptibility, which will be the dominant contribution:

unpaired(T) (T)T

(1.26)

where (T) is the number of unpaired spins at temperature T.The function (T) depends on the distribution of exchange couplings,

P(J). Let us suppose that there is a small density n of randomly-situatedlocalized spins, and suppose that the coupling between spins separated by adistance r is J(r) = J0 e2r/a for some constant a which is roughly the size

of the localized states. With this functional form for J, we know P(J) if weknow P(R) such that P(R)dR is the probability that a spin has its nearestneighbor lying in a thin shell between R and R + dR. We can computeP(R) in the following way. Define p(R) as the probability that the nearestneighbor is a distance R or less. Then

P(R) =dp

dR(1.27)

The probability that the nearest neighbor is between R and R + dR is givenby the number of impurities in a thin shell at R, 4nR2dR, multiplied bythe probability that there is no closer neighbor, 1 p(R)

P(R) = 4nR2(1 p(R)) (1.28)

or, simply,dp

dR= 4nR2(1 p) (1.29)


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Solving this differential equation, we have

p(R) = 1 e4nR3 P(R) = 4nR2 e4nR3 (1.30)We substitute this expression into (1.25),

(T) =1

T

dR 4nR2 e

4

3nR3 2

3 + eJ(R)(1.31)

In the low-temperature limit, this is simply

(T) 1T

dR 8nR2 e

4

3nR3 eJ0 exp(2R/a) (1.32)

The integrand has a saddle-point at R a2 ln(J0/T), dropping ln ln T termswhich are subleading at low T. Hence, the integral can be evaluated bysaddle-point approximation:

(T) 1T

e1

6na3 ln3(J0/T) (1.33)

The physical interpretation of this result is simple. Unpaired spins at tem-perature T are those spins whose largest coupling to the other spins is lessthan T:

J(R) T R a2

ln (J0/T) (1.34)

The density of such spins should therefore be:

(T) e 1

6

na3 ln3(J0/T)

(1.35)

as we found above.The approximate form which we used for P(R) underestimates the num-

ber of unpaired spins, thereby underestimating the susceptibility. We tookP(R) to be the probability of having a neighboring spin at R, namely 4nR2,multiplied by the probability that there were no closer neighbors, 1 p(R).However, the neighboring spin at R might itself have already paired up witha third spin, in which case our spin must look for an even further neighborwith which to form a singlet pair. Thus, we should multiply by the prob-ability that the neighbor at distance R doesnt have a closer neighbor ofits own, which is roughly 1

p(R). (It should really be a fraction of this

because we should exclude the region which is less than a distance R fromboth spins since this region has already been accounted for by the first factorof 1 p(R).) Hence,

P(R) = 4nR2(1 p(R))2 (1.36)


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or

P(R) =4nR2

1 + 43nR32 (1.37)

Substituting this into (1.25), we find the low-temperature form

(T) 1T

dR

8nR21 + 43nR

32 eJ0 exp(2R/a) (1.38)

The saddle-point is again at R a2 ln(J0/T), up to lnln T terms. Hence,the saddle-point approximation for the integral is:

(T) 1T ln3 (J

0/T)

(1.39)

Hence, the number of unpaired spins,

(T) 1ln3 (J0/T)

(1.40)

goes to zero very slowly; so slowly, in fact, that the susceptibility divergesat T 0.

In 1D, the asymptotic low-temperature behavior can be found exactly(refs.).

Exercise: Compute the specific heat at low T in a random singlet phase.

1.5 Physics of the Metallic State

1.5.1 Disorder-Averaged Perturbation Theory

A metallic state in a disordered system can be approached perturbativelyfrom the clean metallic state. In order to do so, we will need to handlethe perturbation caused by impurity scattering. In this section, we describethe simplest way to do so. Let us imagine breaking our system into a largenumber N cells, each of which is macroscopic. Then the configuration ofimpurities in each cell will be different. If we compute an extensive quantity,such as the free energy, then it will be essentially (up to boundary terms)

the sum of the free energies of each of the cells. Since each cell will havea different impurity configuration, the sum over cells will be an averageover impurity configurations. Hence, the free energy in a disordered systemcan be computed by computing the free energy for a fixed realization of thedisorder and then averaging over all realizations of the disorder (with respect


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1.5. PHYSICS OF THE METALLIC STATE 17

(b)

k

V(q)

k+qk k+q

k k q

n v2i

(a)

Figure 1.3: (a) The impurity vertex for a fixed V(q). (b) The inducedinteraction which results from averaging over the propability distributionfor V(q).

to some probability distribution for the disorder). Such an approximation(which we take to be exact in the thermodynamic limit) for the free energy iscalled the quenched approximation and such an average is called a quenchedaverage.

Let us consider a perturbative computation of the free energy. For thesake of concreteness, suppose that the action is

S[,

] = Sclean[,

]] +

d d3

x V(x)

(x, )(x, ) (1.41)

Then there is a vertex in the theory of the form depicted in figure 1.3a.To each such vertex, we assign a factor of the Fourier transform of thepotential, V(q) and integrate over q. We can therefore organize the freeenergy in powers of V(q):

F [V(q)] =n

1

n!

. . .

V(q1) . . . V (qn)f(q1, . . . qn) (1.42)

If we think about computing the free energy p erturbatively in somecoupling constant in Sclean (which includes the simplest case, namely thatSclean is a free theory), then the free energy will only contain connecteddiagrams.

Now suppose that we average over V(q). Let us consider, for simplicity,a distribution for V(q) of the form

V(q) = 0


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V(q)V(q) = niv2(q + q) (1.43)

where ni is the impurity concentration and v is a measure of the strength ofthe scattering potential due to each impurity. All higher moments are takento be factorizable into products of V V. In other words, we will averageF [V(q)] over the distribution:

F =

DV F[V] e

dxV2(x)/2v2 (1.44)

Integrating over V ties together different impurity vertices with impuritylines in all possible ways, according to Wicks theorem. To each such line,we attach a factor of niv

2(q + q), as shown in figure 1.3b.In essence, we have a new Feynman rule associated with impurity lines

connecting impurity vertices. There are two important differences with mostof the Feynman rules which we have considered thus far. (1) No energy flowsalong these lines, only momentum. (2) The diagrams must be connectedeven if all of the impurity lines are cut. In particular, if the clean theoryis a non-interacting theory, then the disorder-averaged perturbation theorycan have no closed electron loops. For example, figure 1.4a is allowed, but1.4b is not.

1.5.2 Lifetime, Mean-Free-Path

The properties of a dirty metal can be derived neatly with the use of an

effective field theory which we will discuss in the next chapter. Therefore,we will just summarize these properties here and describe their qualitativefeatures.

Many of the properties of a dirty metal are qualitatively the same asthose of a clean metal: the compressibility is finite and approximatelytemperature-independent at low-temperatures, as is the magnetic suscep-tibility. The specific heat is linear in T. The principal new feature in a dirtymetal is that the DC conductivity does not diverge at low-temperatures; itapproaches a constant value at T = 0.

This can be seen with a perturbative calculation. Consider the electronGreen function. The lowest-order contribution to the self-energy comes fromthe diagram depicted in figure 1.4a. We will call the imaginary part ofthis diagram (i/2)sgn(n), for reasons which will be clear shortly. If theinteraction between the electrons and the impurities is

Himp =

d3k

(2)3d3k

(2)3d

2V(k k) (k, ) (k, ) (1.45)


23/44


(b)(a)

Figure 1.4: (a) An allowed diagram in disorder-averaged perturbation the-ory. This diagram is the lowest-order contribution to the self-energy. (b) Adisallowed diargam in disorder-averaged perturbation theory.

where, for simplicity, we take

V(q) = 0 , V(q) V(q) = niv2 (3)(q + q) (1.46)

The disorder-averaged value of this diagram is equal to

1

2sgn(n) Im(, k)

= Im

d3k(2)3

V(k k) V(k k)in k

= 2NF niv

2 sgn(n) (1.47)

The real part merely renormalizes the chemical potential.If we add this self-energy to the inverse Green function, we have a Mat-

subara Green function of the form:

G(k, n) = 1in k + i2sgn(n)

(1.48)

The corresponding retarded and advanced Green functions are:

Gret,adv(k, ) =1

k i2(1.49)

In the presence of impurities, momentum eignenstates are no longer en-ergy eigenstates. The inverse-lifetime 1/ is the energy-uncertainty of a


24/44


(q, ) j(q,)

Figure 1.5: The basic conductivity bubble. The electron lines representlines dressed by the one-loop self-energy correction computed in the previoussection.

momentum eigenstate. Writing k = vF|k kF|, we see that the mean-free-path = vF is the momentum uncertainty of an energy eigenstate. It is

essentially the average distance between scatterings of an electron since themomentum is the inverse of the spatial rate of change of the wavefunctionand the latter is scrambled at collisions.

Fourier transforming the fermion Green function at the Fermi energyinto real space, we find

Gret(x, ) e|x|/ (1.50)Thus, the single-particle Green function decays exponentially, unlike in aclean system, where it has power-law behavior.

1.5.3 Conductivity

With this single-particle Green function, we can perturbatively compute theconductivity. Consider the basic bubble diagram, figure 1.5, at q = 0 butwith each electron line representing the Green function (1.48) rather thanthe bare Green function.

j(0, n)j(0, n) =1

3v2FNF

dk

1

n

1

in + im k + i2n+m1

in k + i2n(1.51)

where we have introduced the shorthand n = sgn(n). The factor of 1/3comes from the angular average over the Fermi surface. As usual, we convertthe sum over Matsubara frequencies to an integral

j(0, n)j(0, n) =

v2F

3NF

dk

c

dz

2inF(z)

1

z + in k + i2z+m1

z k + i2z(1.52)


25/44


The integrand is non-analytic if z or z + im is real so the integral is equal

to the contributions from the contour as is passes just above and just belowthese lines. Let us assume that n > 0.

j(0, n)j(0, n) =

v2F

3NF

dk

d

2inF()

1

+ in k + i2

1

k + i2 1

k i2

v2F

3NF

dk

d

2inF()

1

k + i2 1

k i2

1

in k i2(1.53)

Only the second term in parenthesis in the first line of (1.53) contributes tothe imaginary part of the integral and only the first term in parenthesis inthe second line of (1.53) contributes to the imaginary part of the integralbecause the other terms have poles on the same side of the axis. Hence, wehave

j(0, n)j(0, n) =v2F3 NF

dk

d

2i nF()

1

+ in k + i21

k i2 v

2F

3NF

dk

d

2inF()

1

k + i21

in k i2(1.54)

Taking in and dividing by , we have

() =

1

v2F

3

NFdk

d

2

nF()1

+ k +i2

1

k i2

1

v2F3

NF

dk

d

2nF()

1

k + i21

k i2(1.55)


26/44


Shifting + in the second integral and combining the two terms, wehave:

() =

1

v2F3

NF

dk

d

2(nF() nF( + )) 1

+ k + i21

k i2

(1.56)

Taking 0 and noting that (nF() nF( + )) / (), we have

DC

=v2F

6N

Fd

k

1

2k +12

2

=1

3v2F NF

132

k2F (1.57)

As expected, it is a constant determined by the lifetime due to impurityscattering. In the final line, we have written vFNF = k

2F/

2. (This is the3D expression. In general, it is proportional to kd1F .)

The finite conductivity of electrons reflects the fact that electrons movediffusively rather than ballistically. If we define a diffusion constant D =13 v

2F, then = NFD, which is the Einstein relation. Further insight into

this can be gained by computing the density-density correlation function, towhich we turn in the next section.

Note that we have not included vertex corrections in our calculation.This is not a problem because there are no vertex corrections to the currentvertex to this order, as may be seen from the Ward identity, which relatesvertex corrections to the derivative of the self-energy.

(p, p, 0) =(p)

p(1.58)

As usual, we have used the notation p = (, pi), i = 1, 2, 3. Since the self-energy is momentum-independent, there are no vertex corrections to the

current vertices 0

. On the other hand, the self-energy is energy-dependent,so there are important vertex corrections to the density vertex, 0.

Note that we computed this integral by performing the energy integralsfirst and then the momentum integrals. When these integrations do not com-mute, e.g. when the integrands are formally divergent by power-counting,


27/44


it is safest to introduce a momentum cutoff. When this is done, the order

of integrations does not matter. A contour integration over k cannot bedone, however, as a result of the cutoff, so the most natural way to proceedis to perform the energy integral first. The resulting momentum integralis then convergent in most cases. When the integrals are convergent, itdoesnt matter in what order they are done. When they do not converge,as in the present case, the order does matter. The order in which we didthe integrals is safest since it can be done with a momentum cutoff. If wehad ignored this subtlety and done the integrals in the opposite order, anddone the momentum integral without a cutoff (e.g. by contour integration)then we would have actually obtained the same result for the real part of thecurrent-current correlation function. However, the imaginary part, whichincludes a term which cancels the diamagnetic term, would be missing.

1.5.4 Diffusion

The finite diffusion constant of electrons, D, which we found indirectly in theDC conductivity above, can be obtained explicitly by computing the density-density correlation function. As we will see, this correlation function has avery different form in a disordered electron system.

Consider the set of diagrams in figure (1.6)a (with, again, the electronGreen function dressed with the one-loop self-energy, i.e. the lifetime). All of

these diagrams contribute to the density-density correlation function. Theyform a geometric series which we can sum once we have obtained the valueof the first. The higher-order diagrams constitute vertex corrections to thefirst diagram in the series. As noted in the previous subsection, there areno vertex corrections to the conductivity (i.e. the corresponding diagramsfor the current-current correlation function do not correct the vertex in theDC limit, which is why we computed only a single diagram). However, suchvertex corrections are important for the density-density correlation function.We will show that the resulting correlation function is consistent with theWard identity.

The sum of these diagrams is

(q, m) (q, m) = 1

n

d3k

(2)3(q, n, n+m) G(k, n) G(k+q, n+m)

(1.59)


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k+q

+ +++ ...(a)

(b) 1 + +

k +q

kk k

k +qk+q

+...

k k

k +q

Figure 1.6: (a) A geometric series of diagrams which contribute to thedensity-density correlation function. (b) The quantity (q, n, n+m) whichis the ladder sum which sits inside the density-density bubble.

where (q, n, n + m) is the infinite series of ladder diagrams in (1.6)b

(q, n, n + m) = 1 + niv2I(q, n, n + m)

+

niv22

(I(q, n, n + m))2 + . . . (1.60)

and the integral I(q, n, n + m) is given by

I(q, n, n + m) = d3k(2)3G

(k, n)G

(k + q, n + m) (1.61)

Written explicitly,

I(q, n, n + m) =

NF2

dk d(cos )

1

in + im k vFq cos + i2n+m1

in k + i2n(1.62)

where we have used the shorthand n = sgn(n). Integrating k, we have:

I(q, n, n + m) =

iNF

d(cos )

(n) (n + m)im vFq cos + i [(n + m) (n)]

(1.63)


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Integrating cos ,

I(q, n, n + m) =

iNFvFq

[(n) (n + m)]ln

im vFq + i [(n + m) (n)]im + vFq +

i [(n + m) (n)]

=iNFvFq

[(n) (n + m)]ln

1 + (m + ivF q) [(n + m) (n)]1 + (m ivF q) [(n + m) (n)]

(1.64)

Expanding to lowest non-trivial order in m and q, we have

I(q, n, n + m) = iNFvFq

[(n + m) (n)]2

2ivF q 2ivF q m[(n + m) (n)] + 23

(ivF q)3

(1.65)

This expression vanishes unless n, n+m have opposite signs, in which case:

I(q, n(n + m) < 0) = 2NF

1 |m| 1

3v2F

2q2

= 2NF

1 |m| D q2

(1.66)

where the diffusion constant D is given by D = v2F /3.Summing the geometric series (1.60),

(q, n, n + m) =1

1

niv2I(q, n, n + m)

= 11 (1 |m| D q2) [(n + m) (n)]2

=1

1

|m| + Dq2 (nn+m) + (nn+m) (1.67)

where we have used 2NF niv2 = 1. The same form is obtained in any

dimension, with minor differences, such as D = v2F /d in d dimensions.This form indicates that the electron density diffuses.

To make this a little more concrete, consider the density-density corre-lation function (1.59):

(q, m) (

q,

m)

=

1

n d

3k

(2)3

(nn+m)

G(k, n)

G(k+q, n+m)

+1

n

d3k

(2)31

1

|m| + Dq2 (nn+m) G(k, n) G(k + q, n + m)

(1.68)


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The first term is essentially the static response of free fermions with = up to corrections of order 1/, m:

1

n

d3k

(2)3(nn+m) G(k, n) G(k + q, n + m) =

1

n

d3k

(2)31

in k vFq cos 1

in k + O(1/,m)

= NF + O(1/,m) (1.69)In order to do this integral correctly, we must integrate frequency first orelse work with a cutoff in momentum space. Integrating in the oppositeorder will give us an answer which vanishes in the m = 0 limit, i.e. we will

miss the static part. By separating the density-density correlation functioninto static and dynamic pieces the two terms of (1.68) we are left with adynamic piece which is strongly convergent because the Matusbara sum isover a finite range. Hence, we can do the integrals in the opposite order inevaluating that term, to which we now turn.

The second term can be evaluated by doing a k contour integral:

1

n

d3k

(2)31

1

|m| + Dq2 (nn+m) G(k, n) G(k + q, n + m) =

2i

NF2 n

1

1

|m

|+ Dq2

[(n+m) (n)]im

vFq cos +

i [(n+m)

(n)]

2i

NF2

n

1

1

|m| + Dq2(nn+m)

i

= NF|m|

|m| + Dq2 (1.70)

Combining the two terms, we have

(q, m) (q, m) = NF Dq2

|m| + Dq2 (1.71)

This correlation function has a pole at |m| = Dq2. In order to continueto real time, observe that f(z) =

iz sgn(Im(z))) =

|m

|if z = im. If we

continue to z = + i, then we have f( + i) = i, so that the retardeddensity-density correlation function becomes:

(q, ) (q, ) = NF

Dq2

i + Dq2

(1.72)


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The existence of a pole at i = Dq2 means that the Fourier transform hasthe form: [(x, t), (0, 0)] (t) 1

td/2+1e

x2

2Dt (t) (1.73)

Hence, the electron density moves diffusively since a small, highly localizedwavepacket prepared at t = 0 will spread out over a region of size x t1/2at time t.

We obtained a diffusive form for the propagation of electron density bysumming a particular class of diagrams, but these are clearly not the onlypossible diagrams. However, since the density-density correlation functionwhich we obtained vanishes at q = 0, as required by the Ward identity, ourneglect of other diagrams is at least consistent with charge conservation.There is a further justification for neglecting other diagrams: they are sup-

pressed by a power of 1/kF. However, we will find in the next section thatsuch an argument is too quick.

With the density-density correlation function (1.72) in hand, we canre-visit the conductivity. Charge conservation tells us that

t + j = 0 (1.74)Fourier transforming this equation, we have the following relation betweencorrelation functions:

q2 j(q, )j(q, ) = 2 (q, ) (q, ) (1.75)Hence, from the previous expression for the density-density correlation func-

tion, we have 1

ij(q, )j(q, ) = NF i D

i + Dq2(1.76)

Taking q = 0, and then 0, we have the DC conductivity.The spin-spin correlation function also has a diffusive form. Consider,

for instance, the Sz Sz correlation function. For non-interacting electrons,it is precisely the same as the density-density correlation function becauseSz = implies that SzSz = + = (the cross-termsvanish for non-interacting electrons). by rotational symmetry, the otherspin-spin correlation functions must be identical. This can also be easilyseen by direct calculation of S+S: the electron line will have spin +1/2

while the hole line will have spin 1/2, but the diagrams will be unchangedfrom the density-density calculation because the fermion propagators arespin-independent:

S+(q, ) S(q, ) = NF

Dq2

i + Dq2

(1.77)


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In the presence of impurities, electrons no longer move ballistically at

long length scales. They move diffusively b ecause they undergo many col-lisions with impurities. We can think of this in the following RG language.Consider the effective Lagrangian for a Fermi liquid in the presence of im-purities.

S =kd1F(2)d

dl dd

d

2 (i vFl)

+

d3k

(2)3d3k

(2)3d

2V(k k) (k, ) (k, ) (1.78)

To avoid clutter, we suppress the marginal four-Fermi interactions parametrizedby Landau parameters which are inessential to this discussion.

For a fixed V(k

), this term is precisely marginal. This is what wouldbe expected from a single-impurity. Consider, now, a random V(k). Whenwe integrate over different impurity configurations according to (1.46) wegenerate the effective action

Seff =kd1F(2)d

dl dd

d

2 (i vFl)

+ niv2

d3k

(2)3d3k

(2)3d

2

d3p

(2)3d3p

(2)3d

2(k, ) (k, )(p, ) (p, )

(1.79)

with the proviso that we should only allow diagrams which would remainconnected even if all of the impurity lines (connecting the two pairs offermions in the second line in equation (1.79)) were cut. (We will intro-duce a formal way of implementing this in the next chapter.) The resultingimpurity-scattering term is strongly-relevant, scaling as s1. (It is similarto a four-Fermi interaction but with one fewer energy integration.) Thusthe clean Fermi liquid is unstable to impurity-scattering. The RG flow istowards the diffusive Fermi liquid. At this fixed point, the density-densityand spin-spin correlation functions takes the diffusive form (1.72), (1.77)while the single-fermion Green function (1.48) is short-ranged as a resultof the i/ in the denominator. In the next chapter, we will construct aneffective field theory for the long-wavelength, low frequency modes of thediffusive fixed point.

1.5.5 Weak Localization

In the previous two subsections, we expended substantial effort to computethe electron Green function, and the density-density and spin-spin correla-


33/44


k+q

+

...+

...+

k q

k

k

k

k

k

k

k

kk q

...+

k+q

k

k+p

p

++(a)

(b)

+

+ +

(c)+

Figure 1.7: (a) Diagrams contributing to the conductivity with impuritylines maximally-crossed. (b) A re-drawing of the maximally-crossed dia-grams which emphasizes the role of the particle-particle diagrams shown in(c).

tion functions. However, the result of all of these diagrammatic calculationswas no more than simple qualitative considerations and naive Drude theorywould have given us: electrons have a finite conductivity which is determined

by the lifetime . This lifetime also enters the diffusive form of various cor-relation functions through the diffusion constant D = v2F /d. One hardlyneeds sophisticated field-theoretic techniques to discover this, although itis, perhaps, some consolation that we can confirm our intuition by directcalculation.

One might naively think that, in the absence of interactions, this is allthat there is to the conductivity of a dirty metal. However, the sub-leadingimpurity contributions to the conductivity are also interesting. They are dueto quantum interference effects. In two dimensions, the subleading correc-tion is divergent at low-temperatures and, in fact, non-interacting electronsare always insulating in d = 2. Field-theoretic techniques are invaluable in

obtaining and understanding these corrections.Consider the diagrams of figure 1.7a. In these diagrams, the impurity

lines are maximally-crossed. This is a particular class of diagrams contribut-ing to the conductivity. In this subsection, we will see why these diagramsgive a significant contribution to the condutivity. Observe that they can


34/44


=

k

k

kk q

k+q

k

k+p

p

k k

k

k+q k+p

kk+q k+p

k

Figure 1.8: A particle-particle diagram is equal to a particle-hole diagramwith particle momenta reversed to become hole momenta.

be re-drawn in the manner shown in figure 1.7b. The diagrams of 1.7b arevery similar to those of the diffusion propagator. The main difference is

that the arrows are in the same direction, rather than opposite directions,i.e. the middle of the diagram is a particle-particle diagram rather than aparticle-hole diagram. However, time-reversal symmetry relates these two.Call the sum of particle-particle diagrams in figure 1.7c W(k, k, n, n+m).By time-reversal symmetry, we can reverse the direction of one of the arrowsif we also reverse the sign of the momentum. Because particles and holesboth cost positive energy, but carry opposite momentum, a hole line runningthrough a diagram can be replaced with a particle line at the same energiesbut opposite momenta without changing the value of the diagram. This isdepicted in figure 1.8. From this observation, we see that:

W(k, k

, n, n+m) =

niv22

I(k + k

, n, n+m)+

niv23

I(k + k, n, n+m)2

+ . . .

=

niv

22

I(k + k, n, n+m)

1 (niv2)I(k + k, n, n+m)=

niv2

1

|m| + D(k + k)2

(nn+m) (1.80)

If we substitute this into a conductivity bubble to compute its contribu-tion to the conductivity, as in figure 1.7b, we obtain

(m) =1

m

v2F

d

ddk

(2)d

ddk

(2)d

1

n G

(k, n)

G(k, n + m)

W(k, k, n, n+m) G(k, n) G(k, n + m) (1.81)

From (1.80), we see that W(k, k, n, n+m) is sharply peaked at k k.Hence, we write q = k + k and change the variables of integration to k, q.


35/44


In the second pair of Green functions, we take k = k, neglecting the weakq dependence of those factors. We now have:

(m) =1

mNF

v2Fd

dk

ddq

(2)d1

n

niv2

(nn+m)|m| + Dq2

1

in k + i2n

21

in + im k + i2n+m

2

=4i

mNF

v2Fd

ddq

(2)d1

n

niv2

(n+m) (n)

|m| + Dq2

1im +

i((n+m) (n))

3

4im

NFv2Fd

ddq

(2)d1

n

niv2

(n+m) (n)

|m| + Dq2

1i((n+m) (n))

3

= 4m

NFv2Fd

3

ddq

(2)d1

n

niv2

(n+mn)|m| + Dq2

= 1

D

ddq

(2)d1

|m

|+ Dq2

= 1

2d/2

(d2 ) (2)d

1

d 2

1

d2 |m|

D

d22

(1.82)

We have taken the ultraviolet cutoff to be 1/. For shorter wavelengths, oneshould use ballistic, rather than diffusive propagators.

While (1.81) is a small correction in the low frequency limit for d > 2,it is divergent in this limit for d < 2. In d = 2, it is;

(m) = 142

ln

D

|m| 2

(m)

0 = 1

kF ln D|m| 2 (1.83)

which is also divergent.

Hence, in d 2, the correction (1.81)-(1.83), which is formally smallerthan the (semiclassical) Boltzmann result by a power of kF D is, in


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fact divergent as 0. Thus, the starting point of the diffusive fixedpoint is unstable. In the next section, we will discuss the result of thisinstability in more detail but, for now, we observe that it suppresses theconductivity. Thus, it drives the system towards localization. In d = 2,when this correction is still weak (sufficiently high frequencies or, as we willdescribe below, temperatures) and growing only logarithmically, it is calledweak localization.

One might wonder whether is is valid to stop with the diagrams which wehave calculated. Isnt it possible that if we computed some more diagrams,we would find other divergent corrections to the conductivity which mightchange our conclusions entirely? The answer is that this will not happen andwe can stop here. The divergences which we find are due to the existenceof slow modes in the system. The diffusive mode is a slow mode because itis related to the conservation of charge. The particle-particle mode (calleda Cooperon) is slow because it is related to the diffusion mode by time-reversal symmetry. There is no other mode which is guaranteed to be slow bysymmetry, so we do not need to worry about other entirely new divergences,although these same slow modes can lead to subleading divergences. Wewill discuss this point in the next chapter using an effective field theory forthe slow modes of the system.

Thus far, we have considered non-interacting electrons and neglectedphonons, etc. If these sources of inelastic scatteringare considered, they cutoff the above divergences since they break the symmetry which allowed usto substitute a hole line for a particle line. For length scales longer than the

Thouless lengthThouless =

Din (1.84)

inelastic processes will thwart localization and cause ohmic conduction. Forexample, the 2D weak-localization correction is:

(m) = 12NF2

1

2Dln(Thouless/) (1.85)

If in Tp, then the frequency is replaced by Tp, not T, as one mightnaively assume.

A weak magnetic field will also cut off localization at length scales longer

than the magnetic length H = 1H. (Restoring the fundamental constants,it is H =

c/eH.) Thus, in the weak localization regime, where

(m) = 12NF2

1

2Dln(H/) (1.86)


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1.6. THE METAL-INSULATOR TRANSITION 33

the conductivity increases as the magnetic field is increased, i.e. the mag-

netoresistance is negative.

1.5.6 Quasiclassical Approach to Weak Localization***

1.5.7 Weak Magnetic Fields and Spin-orbit Interactions: theUnitary and Symplectic Ensembles

1.6 The Metal-Insulator Transition

1.6.1 Percolation

Consider a disordered solid which is insulating as a result of impurities.

Although it is insulating, there will be some isolated regions which are con-ducting. In a system of non-interacting fermions, such regions would havesingle-particle states at the Fermi energy which extend across them but donot leak substantially into the surrounding insulating regions. One way tothink about this is to consider the potential V(x), whose spatial variationis random as a result of impurities. The regions in which V(x) < F areclassically-allowed for electrons at the Fermi energy. Let us consider these tobe the conducting regions. The regions in which V(x) > F are classically-forbidden for electrons at (or b elow) the Fermi energy. Let us call theseinsulating regions.

Suppose we decrease the impurity concentration. Then the conducting

regions will grow in size. As we continue to decrease the impurity concen-tration, the conducting regions will grow larger and larger and will begin tomerge. Eventually, it will become possible to go from one end of the systemto the other within one long conducting region which spans the system. Ifelectrical conduction were purely classical and ohmic, this point whichis called the percolation threshold would be the transition point betweenmetallic and insulating behavior.

However, electrical conduction is not classical. Even below the perco-lation threshold, electrons can tunnel quantum mechanically from one con-ducting region to another. Conversely, an electron can take two differentpaths from point A to point B, and the quantum mechanical amplitudes for

these processes might interfere destructively. Nevertheless, at temperatureswhich are higher than the characteristic phase coherence temperatures ofthe system so that coherent quantum tunneling cannot occur but not sohigh that the conductivity of the insulating regions (insulating and metal-lic are precise distinctions only at zero temperature; at finite temperatures,


38/44


all systems have finite conductivity) is comparable to that of the conduct-

ing ones, we expect the conductivity and other physical properties to showbehavior characteristic of percolation processes. Since percolation is, in asense, the classical limit of the metal-insulator transition, we will discuss itbriefly.

There are two basic lattice models of percolation, site percolation andbond percolation. In site percolation, one removes sites randomly from alattice and considers the properties of the remaining sites. We will say thata site is present if it has not been removed and absent if it has been. Weimagine that an electron can hop directly from a given site which is presentto its nearest neighbors if they are present, but not to other sites. Thus, if anthere is an isolated present site which is surrounded by absent sites, then anelectron on that site is stuck there. Sites which are present can be groupedinto clusters of sites which are continguous with each other. One site in acluster can be reached from any other site in the cluster by a sequence ofnearest-neighbor hops entirely within the cluster. In bond percolation, oneremoves b onds from a lattice. An electron can hop from one lattice site toa nearest-neighbor site only if the bond between them is present. Sites canagain be grouped into clusters which are connected by bonds so that anysite in cluster can be reached from any other site in the cluster.

For the sake of concreteness, let us consider bond percolation. Whenthe average bond density, which we will call p, is low, clusters tend to besmall. As the bond density is increased, clusters grow until the percolationthreshold, pc, is reached. At this point, one cluster extends across the length

of the system; it is called the percolating cluster. For p < pc, the averagedistance between two sites on the same cluster, , scales as

(pc p) (1.87)

For p > pc, the fraction, P, of the system which belongs to the percolatingcluster is

P(p) (p pc) (1.88)(For p < pc, P = 0 because the largest cluster in an infinite system is finite.)Notice the analogy between these exponents and the correlation length and

order-parameter exponents associated with critical phenomena.One might imagine that the conductivity (assuming that bonds which

are present are considered to be conducting and those which are absent areinsulating) would be proportional to the fractions of the system belongingto the largest cluster. However, this is not the case because many of the


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bonds in the cluster lead to dead-ends which do not contribute to the con-

ductivity. Instead, the conductivity is associated with a different exponent,, (sometimes called t) which describes the backbone of the percolatingcluster i.e. the cluster with dead-ends (those bonds which do not have twocompletely independent routes to the two ends of the cluster) removed.

(p) (p pc) (1.89)At the percolation point itself, the system will have power-law correla-

tions. Many of the properties of the percolation p oint in two dimensionshave been revealed using conformal field theory and, more recently with theuse of the stochastic Loewner equation.

However, as we pointed out earlier, percolation does not correctly de-

scribe metal-insulator transitions because it misses quantum-mechanical ef-fects. In the next section, we discuss alternate approaches which take theseeffects into account.

1.6.2 Mobility Edge, Minimum Metallic Conductivity

Let us now consider a metal-insulator transition in a system of non-interactingfermions. Such a transition can occur as a result of varying either the chem-ical potential or the disorder strength. If the system is weakly-disordered,then the energy spectrum will still approximately break up into bands, al-though there may not be a true gap between the bands because there will beexponentially-small tails in the density of states. Near the bottom of a band,

the states will all be localized because the kinetic energies of these stateswill be low. Near the top of the band, the same will be true. It is useful tothink of a band with chemical potential near the top as a nearly empty holeband; in such a case, the hole kinetic energies near are low near the Fermienergy. Near the middle of the band, on the other hand, the states willbe extended if the disorder strength is sufficiently small because the kineticenergies will be large in comparison. (If the disorder strength is too large,then even at the middle of the band the kinetic energy will not be largeenough to overcome it.) What happens as the Fermi energy is swept fromthe bottom of the band to the middle? To answer this question, one shouldfirst note that there are no energies at which there are both extended and

localized states because localization is not stable in the presence of extendedstates at the same energy. A small change in the particular realization ofdisorder (e.g. the locations of the impurities) would cause mixing betweenthe localized states and the extended ones. When a localized state mixeswith extended states, it becomes extended.


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states

mobility

E

(E)

Ec extendedstates

localized

edge

Figure 1.9: The density-of-states in a disordered system, with extended andlocalized states and mobility edges shown.

Hence, there will be sharp energies which separate localized states fromextended ones. Such an energy is called a mobility edge. As depicted infigure 1.9, the mobility edge separates the strongly-localized states with lowkinetic energy in the band tails from the extended states with high kineticenergy in the center of the band. (One can imagine there being more than asingle pair of such energies, but this is not generic and would require somespecial circumstances.)

As the chemical potential is moved through the mobility edge, Ec, the

system undergoes a metal-insulator transition. For EF < Ec, the zero-temperature DC conductivity vanishes. For EF > Ec, it is finite. Whathappens at EF = Ec? Is it equal to some minimum non-zero value min,the minimum metallic conductivity? According to the percolation modeldescribed in the preceding section, the conductivity vanishes precisely atthe transition, |EF Ec|. (EF Ec or any other control parameterwill act as a proxy for p pc near the transition.) On the other hand, thefollowing heuristic argument suggests that it is finite. The conductivity isgiven in Boltzmann transport theory by

= Ade2

kd1F (1.90)

where is the mean-free-path and Ad is a dimension-dependent constant.(Here, we have restored the factors of e, which were set to one earlier.) Ifscattering is due entirely to impurities, the mean-free path cannot be shorterthan the inter-atomic spacing a in the solid. Hence, kF

> since kF /a.


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Hence, we conclude that

min = Ade2

kd2F (1.91)

In particular, in d = 2, the minimum metallic conductivity can be writtenentirely in terms of fundamental constants, min = Ade

2/ (restoring thefundamental constants which we have set to 1).

However, one should be cautious in applying the Boltzmann transportformula (1.90) which is semiclassical and valid for large kF in the regimekF 1. Indeed, as we will see in the next subsection, the preceding logicfails in the case of non-interacting electrons.

1.6.3 Scaling Theory

Thouless had the important insight that one should study the scaling be-havior of the conductance. Consider a system of non-interacting electronsof size L. Thouless wrote the conductance of the system in the followingform, which we encountered earlier (1.6):

g(L) =E

E(1.92)

E is the disorder-avergaged energy change of eigenstates at EF when theboundary conditions are changed from periodic to antiperiodic. E is themean level spacing of eigenstates. (In order to derive such a relation ana-

lytically, one must consider disorder-averaged quantities.)To understand this formula, suppose that D(L) is the diffusion constantfor an electron at the Fermi energy. The time required for an electron todiffuse across the system is the Thouless time:

tL =L2

D(L)(1.93)

According to the uncertainty principle,

E 1tL

D(L)L2

(1.94)

Meanwhile,

E = 1NF Ld

(1.95)

Hence,E

E= NF D(L) L

d2 = Ld2 = g(L) (1.96)


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The second equality follows from the Einstein relation between the diffusion

constant and the conductivity.We now study how g(L) scales as the system size is increased. We couldincrease the system size from L to bL by stacking b subsystems of size L. Thekey assumption of Abrahams, Anderson, Licciardello, and Ramakrishnan(the gang of four) is that:

g(bL) = f(b, g(L)) (1.97)

In particular, L does not appear explicitly on the right-hand-side. In amacroscopic ohmic system, such a relation is obvious, g(bL) = bd2g(L).However, this is not exactly right microscopically. While E(bL) = bdE(L)should hold generally, we need to find a relation for E(bL). This relation

might be complicated since we have to match the eigenfunctions from eachsize L block at their boudaries in order to construct the eigenfunctions ofthe size bL system. However, the sensitivity of the eigenfunctions to theirboundary conditions is precisely what is encapsulated in g(L). Thus, it isvery reasonable to assume that g(bL) depends only on b and g(L). As notedabove, this assumption holds in the ohmic regime. It also holds in the lo-calized regime, where g(L) = g0 e

L/ ( is the localization length), sinceg(bL) = g0 (g(L)/g0)

b. The scaling assumption of the gang of four was thatthe form (1.97) holds throughout the entire region between these limits.

If (1.97) holds, then

d ln g

d ln L =

1

gdf

db

b=1= (g) (1.98)

In strongly-disordered systems, g is small, and all states are localized. Inthis limit, we expect (g) = ln(g/g0). In the clean, or large g, limit, weexpect Ohmic scaling, and (g) = d 2.

How do we connect the two limits? For g large, we can compute (g)in powers of 1/g or, equivalently, 1/kF. In the previous section, we havealready computed the first correction in 1/kF to the conductivity. If wereplace the inelastic scattering time with the system size as an infraredcutoff, then this expression gives us (L) for large kF. Differentiating, weobtain (g).

(g) = d 2 a

g + . . . (1.99)

where . . . represents O

1/g2

terms. From our computation of the weak-localization correction to the conductivity in d = 2, a = 1/42. This isinterpolated with the small g limit in figure 1.10.


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d=2g

d=3

g

c

Figure 1.10: The -function of Abrahams, et al. which is obtained bysmoothly connecting the Ohmic limit with the localized one.

There are several consequences which follow immediately. (1) There is nometallic state of non-interacting fermions in d 2 (and, therefore, no metal-insulator transition, of course). (2) There is a metal-insulator transition withno minimum metallic conductivity in d > 2. (3) For d = 2 + , the metal-insulator transition occurs ast g 1/, which is in the perturbative regime.The first statement is clear from figure 1.10: since (g) < 0 for all g, theconductance always flows to the localized regime of small g. The secondstatement follows from the positivity of(g) for large g. In order to connect

to the negative (g) localized regime at small g, (g) must pass through zeroat some gc. For g > gc, (g) > 0 and the conductance flows to large values,where it is Ohmic; this is the metallic state. For g < gc, (g) < 0 and theconductance flows to small values; this is the insulating state. For g = gc,the conductance remains constant; this is the transition point. However,if the conductance remains constant as the system size L goes to infinity,the conductivity c = gcL

2d 0. Hence, there is no minimum metallicconductivity.

The third statement follows immediately from (1.99). We can say moreabout the critical region by expanding the -function about gc:

(g) =

1

g gcgc (1.100)

From our -function (1.99), we see that 1/ = and gc = ad/, where = d 2. Since the . . . in (1.99) can be neglected only for g large, we cantrust our gc = ad/ conclusion only for small, i.e. only near two dimensions.


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Suppose |g gc| is small at the microscopic scale . The linearized -function is applicable. Let us integrate until |g gc|/gc 1. Let us definethe length scale at which this is reached as :

=

g gcgc

(1.101)

This is an important length scale because the system will cross over to Ohmicmetallic (g < gc) or localized insulating (g < gc) behavior at longer lengthscales. On the insulating side, is the localization length.

On the conducting side, is a classical-quantum crossover length. Forlength scales longer than , electrical conduction is Ohmic and, therefore,essentially classical. For length scales shorter than , quantum interference

effects can occur and classical ideas are inappropriate. The conductivity isequal to

=gc

(1.102)

The vanishing of the minimum metallic conductivity can be understood asthe divergence of at the metal-insulator transition.

Thus, the scaling hypothesis of Abrahams, et. al is equivalent to theassumption that there is a single length scale, , which becomes large nearthe metal-insulator transition. This is completely analogous to the case ofclassical thermal phase transitions studied in chapter .. or quantum phasetransitions studied in chapter .. However, there is an important difference:

a metal-insulator transition of non-interacting fermions is not a thermo-dynamic phase transition. While the conductivity changes sharply at thetransition, all thermodynamic properties, such as the ground state energy,are perfectly smooth across the transition. The smoothness of the groundstate energy, for instance, follows from the smoothness of the density-of-states across the transition.

Impurities in Solids

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